number of observations. If length(n) > 1, the length
is taken to be the number required.
time
Time where the futures process is evaluated
(relative to now).
ttm
Time to maturity (relative to now).
s0
Either a numeric representing the initial value of the
commodity spot price or an object inheriting from class
schwartz2f.
delta0
Initial value of the convenience yield.
mu
Drift term of commodity spot price.
sigmaS
Diffusion parameter of the spot price process.
kappa
Speed of mean-reversion of the convenience yield process.
alpha
Mean-level of the convenience yield process.
sigmaE
Diffusion parameter of the convenience yield process.
rho
Correlation coefficient between the Brownian motion
driving the spot price and the convenience yield process.
lambda
Market price of convenience yield risk (see
Details).
alphaT
Mean-level of the convenience yield process with
respect to the equivalent martingale measure (see Details).
r
Instantaneous risk-free interest rate.
measure
under which the functions are computed. “P”
denotes the objective measure, “Q” the risk-neutral measure
(see Details).
...
Arguments to be passed to the functions
d/p/q-norm.
Details
Futures prices depend on the spot-price and the convenience yield.
To get the real (i.e. the objective) distribution of futures prices at
some date in the future the dynamics is considered under the objective
measure P. The P-dynamics is
where
W1, W2 are Brownian motions under the objective
measure, the measure P.
Options on futures are evaluated based on the risk-neutral dynamics of
the spot-price and the convenience yield, i.e. under the measure
Q. The Q-dynamics is
where W1*, W2* are
Brownian motions with respect to Q.
alphaT = alpha -
lambda / kappa where lambda is the market price of
convenience-yield risk. The market price of convenience yield risk
can either be specified explicitly by lambda or implicitly by
alphaT. The relation is alphaT = alpha - lambda /
kappa. See the package vignette.
Value
Probabilities, densities, quantiles or samples of the log-normally
distributed futures prices as numeric.
Note
Note that futures and forward prices coincide as the interest rate is
assumed to be constant in the Schwartz two-factor model.
Author(s)
Philipp Erb, David Luethi, Juri Hinz
References
The Stochastic Behavior of Commodity Prices: Implications for
Valuation and Hedging by Eduardo S. Schwartz Journal of Finance
52, 1997, 923-973
Valuation of Commodity Futures and Options under Stochastic
Convenience Yields, Interest Rates, and Jump Diffusions in the Spot
by Jimmy E. Hilliard and Jorge Reis Journal of Financial and
Quantitative Analysis 33, 1998, 61-86
See Also
pricefutures,
d/p/qstate,
r/simstate
Examples
# ## Create a "schwartz2f"-object
# model <- schwartz2f()
#
# ## Probability
# pfutures(q = 10 * 3:9, time = 0.5, ttm = 2, model, lambda = 0.01)
#
# ## Density
# dfutures(x = c(20, 40, 100), time = 0.5, ttm = 2, model, lambda = 0.01)
#
# ## Quantile
# qfutures(p = 0.1 * 2:5, time = 0.5, ttm = 10, model, lambda = 0.01)
#
# ## Sample
# sim <- rfutures(n = 1000, time = 0.5, ttm = 5, model, lambda = 0.01)
#
# hist(sim, prob = TRUE)
# lines(seq(30, 300, length = 100),
# dfutures(seq(30, 300, length = 100),
# time = 0.5, ttm = 5, model, lambda = 0.01), col = "red")
#
# ## At time 0 the futures price is a deterministic function of s0 and
# ## delta0. Therefore 3 times the same value is obtained:
# rfutures(3, time = 0, ttm = 1, model, lambda = 0)